L1 Analyse Développement limité 1 question

exo7 4018

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Fonctions trigonométriques

x/sin⁡x=1+x2/6+7x4/360+o(x4)1/cos⁡x=1+x2/2+5x4/24+o(x4)ln⁡(sin⁡x/x)=−x2/6−x4/180−x6/2835+o(x6)exp⁡(sin⁡x/x)=e(1−x2/6+x4/45)+o(x4)tan⁡x=1+h+h2/2+o(h2),h=x−π/4sin⁡(x+x2+x3−x4)=x+x2+5x3/6−3x4/2+o(x4)ln⁡(xtan⁡(1/x))=x−2/3+7x−4/90+o(1/x4)(1−cos⁡x)/(ex−1)2=1/2−x/2+x2/6+o(x2)sin⁡((πcos⁡x)/2)=1−π2x4/32+π2x6/192+o(x6)cos⁡xln⁡(1+x)=x−x2/2−x3/6+o(x4)(sin⁡x−1)/(cos⁡x+1)=−1/2+x/2−x2/8+o(x2)ln⁡(2cos⁡x+tan⁡x)=ln⁡2+x/2−5x2/8+11x3/24−59x4/192+o(x4)ecos⁡x=e(1−x2/2+x4/6)+o(x5)\begin{align*} x/\sin x &= 1 + x^2/6 + 7x^4/360 + o(x^4) \cr 1/\cos x &= 1 + x^2/2 + 5x^4/24 + o(x^4) \cr \ln(\sin x/x) &= -x^2/6 -x^4/180 -x^6/2835 + o(x^6) \cr \exp(\sin x/x) &= e(1 - x^2/6 + x^4/45) + o(x^4) \cr \sqrt{\tan x} &= 1 + h + h^2/2 + o(h^2), h = x-\pi/4 \cr \sin(x+x^2+x^3-x^4) &= x + x^2 + 5x^3/6 -3x^4/2 + o(x^4) \cr \ln(x\tan(1/x)) &= x^{-2}/3 + 7x^{-4}/90 + o(1/x^4) \cr (1-\cos x)/(e^x-1)^2 &= 1/2 - x/2 + x^2/6 + o(x^2) \cr \sin((\pi\cos x)/2) &=1 -\pi^2x^4/32 + \pi^2x^6/192 + o(x^6) \cr \cos x\ln(1+x) &= x - x^2/2 - x^3/6 + o(x^4) \cr (\sin x-1)/(\cos x+1) &= -1/2 + x/2 - x^2/8 + o(x^2) \cr \ln(2\cos x+\tan x) &= \ln2+x/2-5x^2/8+11x^3/24-59x^4/192 + o(x^4) \cr e^{\cos x} &= e(1 - x^2/2 + x^4/6) + o(x^5) \cr \end{align*}

Fonctions circulaires inverses

arcsin⁡2x=x2+x4/3+8x6/45+o(x6)1/arcsin⁡2x=x−2−1/3−x2/15+o(x2)arctan⁡(x+1)/(x+2)=π/4−x−1/4+3x−2/8arccos⁡(sin⁡x/x)=∣x∣/3(1−x2/90)+o(1/x3)1/arctan⁡x=x−1+x/3−4x3/45+44x5/945+o(x5)arcsin⁡x=π/6+1/3(2h−4h2/3+32h3/9)+o(h3),h=x−1/4arcsin⁡(sin⁡2x)=x2−x4/3+19x6/90−107x8/630+o(x8)arctan⁡(1+x)=π/4+x/2−x2/4+x3/12+o(x4)arcsin⁡x/(x−x2)=1+x+7x2/6+o(x2)earcsin⁡x=eπ/6(1+2h/3+2(1+3)h2/(33))+o(h2),h=x−1/2e1/xarctan⁡x=π2+(π2−1)x−1+(π4−1)x−2+(π12−16)x−3+o(1/x3)\begin{align*} \arcsin^2 x &= x^2 + x^4/3 + 8x^6/45 + o(x^6) \cr 1/\arcsin^2 x &= x^{-2} - 1/3 - x^2/15 + o(x^2) \cr \arctan\sqrt{(x+1)/(x+2)} &= \pi/4 - x^{-1}/4 + 3x^{-2}/8 \cr \arccos(\sin x/x) &= |x|/\sqrt3(1 - x^2/90) + o(1/x^3) \cr 1/\arctan x &= x^{-1} + x/3 -4x^3/45 +44x^5/945 + o(x^5) \cr \arcsin\sqrt x &= \pi/6+1/\sqrt3(2h-4h^2/3+32h^3/9) + o(h^3), h = x-1/4 \cr \arcsin(\sin^2 x) &= x^2 -x^4/3 +19x^6/90 -107x^8/630 + o(x^8) \cr \arctan(1+x) &= \pi/4 + x/2 - x^2/4 + x^3/12 + o(x^4) \cr \arcsin x/(x-x^2) &= 1 + x + 7x^2/6 + o(x^2) \cr e^{\arcsin x} &= e^{\pi/6}(1 + 2h/\sqrt3 + 2(1+\sqrt3)h^2/(3\sqrt3)) + o(h^2), h = x-1/2 \cr e^{1/x}\arctan x &= \frac\pi2+ (\frac\pi2-1)x^{-1} + (\frac\pi4-1)x^{-2} + (\frac\pi{12}-\frac16)x^{-3} + o(1/x^3) \cr \end{align*}

Exponentielle et logarithme

x/(ex−1)=1−x/2+x2/12+o(x2)ln⁡x/x=h−h2+23h3/24+o(h3),h=x−1ln⁡((2−x)/(3−x2))=ln⁡(2/3)−x/2+5x2/24+o(x2)ln⁡(1+x)/(1−x+x2)=x+x2/2−x3/6+o(x3)ch⁡x/ln⁡(1+x)=x−1+1/2+5x/12+o(x)ln⁡(ln⁡(1+x)/x)=−x/2+5x2/24−x3/8+o(x3)ln⁡(ax+bx)=ln⁡2+xln⁡ab+x2ln⁡2(a/b)/8+o(x2)exp⁡(1/x)/x2=e(1−3h+13h2/2−73h3/6)+o(h3),h=x−1\begin{align*} x/(e^x-1) &= 1 - x/2 + x^2/12 + o(x^2) \cr \ln x/\sqrt x &= h - h^2 + 23h^3/24 + o(h^3), h = x-1 \cr \ln((2-x)/(3-x^2)) &= \ln(2/3) - x/2 + 5x^2/24 + o(x^2) \cr \ln(1+x)/(1-x+x^2) &= x + x^2/2 - x^3/6 + o(x^3) \cr \ch x/\ln(1+x) &= x^{-1} + 1/2 + 5x/12 + o(x) \cr \ln(\ln(1+x)/x) &= -x/2 + 5x^2/24 - x^3/8 + o(x^3) \cr \ln(a^x+b^x) &= \ln2 + x\ln\sqrt{ab} + x^2\ln^2(a/b)/8 + o(x^2) \cr \exp(1/x)/x^2 &= e(1 - 3h + 13h^2/2 - 73h^3/6) + o(h^3), h = x-1 \cr \end{align*}

Fonctions hyperboliques inverses

Argth(sin⁡x)=x+x3/6+x5/24+o(x5)Argsh(ex)=ln⁡(1+2)+1/2(x+x2/4)+o(x2)\begin{align*} \Argth(\sin x) &= x + x^3/6 + x^5/24 + o(x^5) \cr \Argsh(e^x) &= \ln(1+\sqrt2) + 1/\sqrt2(x + x^2/4) + o(x^2) \cr \end{align*}

Formes exponentielles

(1−x+x2)1/x=e−1(1+x/2+19x2/24)+o(x2)((1+x)/(1−x))α=1+2αx+2α2x2+2α(2α2+1)x3/3+o(x3)(sin⁡x/x)2/x2=e−1/3(1−x2/90)+o(x3)(sin⁡x/x)3/x2=e−1/2(1−x2/60−139x4/151200)+o(x4)(1+sin⁡x)1/x=e(1−x/2+7x2/24)+o(x2)(1+sin⁡x+cos⁡x)x=1+xln⁡2+x2(ln⁡22+1)/2+o(x2)(sin⁡x)sin⁡x=1−h2/2+7h4/24+o(h4),h=x−π/2(tan⁡x)tan⁡2x=e−1(1+2h2/3+4h4/5)+o(h4),h=x−π/4Deˊvelopper d’abord ln⁡((1+x)/(1−x))\begin{align*} (1-x+x^2)^{1/x} &= e^{-1}(1 + x/2 + 19x^2/24) + o(x^2) \cr ((1+x)/(1-x))^\alpha &= 1 + 2\alpha x + 2\alpha^2x^2 + 2\alpha(2\alpha^2+1)x^3/3 + o(x^3) \cr (\sin x/x)^{2/x^2} &= e^{-1/3}(1 - x^2/90) + o(x^3) \cr (\sin x/x)^{3/x^2} &= e^{-1/2}(1 - x^2/60 - 139x^4/151200)+ o(x^4) \cr (1+\sin x)^{1/x} &= e(1 - x/2 + 7x^2/24) + o(x^2) \cr (1+\sin x + \cos x)^x &= 1 + x\ln2 + x^2(\ln^22+1)/2 + o(x^2) \cr (\sin x)^{\sin x} &= 1 - h^2/2 + 7h^4/24 + o(h^4), h = x-\pi/2 \cr (\tan x)^{\tan2x} &= e^{-1}(1 + 2h^2/3 + 4h^4/5) + o(h^4), h = x-\pi/4 \cr & \text{Développer d'abord $\ln((1+x)/(1-x))$} \cr \end{align*}

Radicaux

x(x−1)/(x+1)=1/3(2+5h/3+h3/54)+o(h3),h=x−21+1−x=2(1−x/8−5x2/128−21x3/1024)+o(x3)1−1−x2=∣x∣/2(1+x2/8+7x4/128)+o(x5)ex−1+2x=x2−x3/3+2x4/3−13x5/15+o(x5)(x3+x23+x3−x23)/x=2−2x−2/9+o(1/x3)\begin{align*} x\sqrt{(x-1)/(x+1)} &= 1/\sqrt3(2 + 5h/3 + h^3/54) + o(h^3), h = x-2 \cr \sqrt{1+\sqrt{1-x}} &= \sqrt2(1 - x/8 - 5x^2/128 - 21x^3/1024) + o(x^3) \cr \sqrt{1-\sqrt{1-x^2}} &= |x|/\sqrt2(1 + x^2/8 + 7x^4/128) + o(x^5) \cr e^x-\sqrt{1+2x} &= x^2 - x^3/3 + 2x^4/3 - 13x^5/15 + o(x^5) \cr (\sqrt[3]{x^3+x^2}+\sqrt[3]{x^3-x^2})/x &= 2 - 2x^{-2}/9 + o(1/x^3) \cr \end{align*}

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